Democrats have so far failed to derail any of President Trump’s nominees, lacking the numbers in the Senate to block them. But they are about to test their ability to use an arcane Senate tradition to stop Mr. Trump from installing one lesser-known but critical category of nominee: federal prosecutors.

For decades, heads of the Senate Judiciary Committee of both parties have sought the approval of home-state senators before moving ahead with the nominations of U.S. attorneys, waiting to proceed until they receive what is known as a “blue slip” indicating the senators’ approval.

Senator Chuck Schumer, the New York Democrat and minority leader, will be the first to challenge Mr. Trump on his selections, by refusing to return blue slips consenting to consider the nominees for two top posts in New York: Jay Clayton to be the U.S. attorney for the Southern District and Joseph Nocella Jr. to be the lead prosecutor in the Eastern District.

“Donald Trump has made clear he has no fidelity to the law and intends to use the Justice Department, the U.S. attorney offices and law enforcement as weapons to go after his perceived enemies,” Mr. Schumer said in a statement to The New York Times confirming his intent to withhold the blue slips. “Such blatant and depraved political motivations are deeply corrosive to the rule of law and leaves me deeply skeptical of Donald Trump’s intentions for these important positions.”

His position is likely to anger the White House. But in a recent interview, Senator Charles E. Grassley, the Iowa Republican who leads the Judiciary Committee, said that he intended to respect the blue slip tradition. Given Mr. Schumer’s stance, that means the two high-priority nominations of Mr. Trump are on track to die in the committee without receiving a vote.

“The answer is yes,” Mr. Grassley said when asked whether he would honor the blue slip position of senators. “If they are from the state the nomination comes from.”